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Theorem bnj1304 31016
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1304.1 (𝜑 → ∃𝑥𝜓)
bnj1304.2 (𝜓𝜒)
bnj1304.3 (𝜓 → ¬ 𝜒)
Assertion
Ref Expression
bnj1304 ¬ 𝜑

Proof of Theorem bnj1304
StepHypRef Expression
1 notnotb 304 . . . 4 (∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒))
2 notnotb 304 . . . . . . . 8 (𝜒 ↔ ¬ ¬ 𝜒)
32anbi2i 730 . . . . . . 7 ((¬ 𝜒𝜒) ↔ (¬ 𝜒 ∧ ¬ ¬ 𝜒))
43exbii 1814 . . . . . 6 (∃𝑥𝜒𝜒) ↔ ∃𝑥𝜒 ∧ ¬ ¬ 𝜒))
5 ioran 510 . . . . . . 7 (¬ (𝜒 ∨ ¬ 𝜒) ↔ (¬ 𝜒 ∧ ¬ ¬ 𝜒))
65exbii 1814 . . . . . 6 (∃𝑥 ¬ (𝜒 ∨ ¬ 𝜒) ↔ ∃𝑥𝜒 ∧ ¬ ¬ 𝜒))
7 exnal 1794 . . . . . 6 (∃𝑥 ¬ (𝜒 ∨ ¬ 𝜒) ↔ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒))
84, 6, 73bitr2ri 289 . . . . 5 (¬ ∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ∃𝑥𝜒𝜒))
98notbii 309 . . . 4 (¬ ¬ ∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ∃𝑥𝜒𝜒))
10 exancom 1827 . . . . 5 (∃𝑥𝜒𝜒) ↔ ∃𝑥(𝜒 ∧ ¬ 𝜒))
1110notbii 309 . . . 4 (¬ ∃𝑥𝜒𝜒) ↔ ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒))
121, 9, 113bitri 286 . . 3 (∀𝑥(𝜒 ∨ ¬ 𝜒) ↔ ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒))
13 exmid 430 . . 3 (𝜒 ∨ ¬ 𝜒)
1412, 13mpgbi 1765 . 2 ¬ ∃𝑥(𝜒 ∧ ¬ 𝜒)
15 bnj1304.1 . . 3 (𝜑 → ∃𝑥𝜓)
16 bnj1304.2 . . . 4 (𝜓𝜒)
17 bnj1304.3 . . . 4 (𝜓 → ¬ 𝜒)
1816, 17jca 553 . . 3 (𝜓 → (𝜒 ∧ ¬ 𝜒))
1915, 18bnj593 30941 . 2 (𝜑 → ∃𝑥(𝜒 ∧ ¬ 𝜒))
2014, 19mto 188 1 ¬ 𝜑
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745 This theorem is referenced by:  bnj1204  31206  bnj1279  31212  bnj1311  31218  bnj1312  31252
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